Locus & Parabola: find the equation of the parabola

Question

Please explain how to solve this question:

Find the equation of the parabola with coordinates of the vertex being $(0,0)$ and equation of the axis $x = 0$, passing through the point $(-1, 7)$.

Thanks

Answer 1

Hint:

So, we can set the equation of the directrix $y=p$

So, the directrix will intersect the axis at $D(0,p)$

As the focus$F(0,c)$ lies on the axis, vertex is the midpoint of $D,F$

$\dfrac{p+c}2=0\iff c=-p$

Now, as the eccentricity$=1$, the distance of $(-1,7)$ from the vertex $=$ the distance from the directrix.

This will give us the possible value(s) of $p$

Now if $P(h,k)$ is any point of the parabola, use the fact that the eccentricity$=1$

Answer 2

A generic parabola with axis parallel to $x = 0$ is of the form $y = ax^2+bx+c$. If $(0,0)$ is one of its points then it must satisfy the equation, hence $$0 = a \cdot0+b \cdot 0+c = c.$$ Further, the first component of the vertex is $-\frac{b}{2a}$, which is $0$, so $b = 0$. Hence $y = ax^2$. Now $(-1,7)$ must satisfy the equation, so $7 = a$, and then $y = 7x^2$.